Complex-step Derivative Research Articles

Hyperelastic Material Modeling Using First- and Second-Order Differential Operations with Complex Step Derivative Approximation

This study proposes a method for calculating hyperelastic materials’ stress and tangent modulus using a generalized complex-step derivative approximation. This method is accurate and requires only the operational code of the strain energy density function. Moreover, because it uses only standard complex operations in Fortran, no prior preparation is required, making the method convenient and practical. The effectiveness of the method was confirmed by applying it to the implementation of isotropic and anisotropic hyperelastic models.

Hyperelastic Model with Energy Limiter for EPDM and Its Numerical Application

To simulate the working process of EPDM PSD of dual-pulse solid rocket motor from beginning to failure, we use a hyperelastic model with erergy limiter to model this working process. In traditional practice, the consistent tangent moduli of mechanical model is derived by analytic method, however, the analytic method is tedious and fallible. To solve this problem, we introduce numerical approximate method (complex-step derivative, which is proved to be of second-order for suitably small perturbation and does not suffer from inherent subtractive cancellations ) to derive the consistent tangent moduli, which is simple, robust, and efficient, and can be easily implemented into commercial finite element software. Finally, we simulate a case of unixial tension of a dumbbell-shape and compare the simulation results with the analytic results, a good agreement between the two kinds of results is observed.

Quaternionic step derivative: Machine precision differentiation of holomorphic functions using complex quaternions

The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them with a small imaginary step next to the real number line. The current paper proposes that derivatives of holomorphic functions can be calculated in a similar fashion by taking a small step in a quaternionic direction instead. It is demonstrated that in so doing the CSD properties of high accuracy and convergence are carried over to derivatives of holomorphic functions. To demonstrate the ease of implementation, numerical experiments were performed using complex quaternions, the geometric algebra of space, and a 2 × 2 matrix representation thereof.

Reliability-Based Design Optimization of Structures Using the Second-Order Reliability Method and Complex-Step Derivative Approximation

This paper proposes a reliability-based design optimization (RBDO) approach that adopts the second-order reliability method (SORM) and complex-step (CS) derivative approximation. The failure probabilities are estimated using the SORM, with Breitung’s formula and the technique established by Hohenbichler and Rackwitz, and their sensitivities are analytically derived. The CS derivative approximation is used to perform the sensitivity analysis based on derivations. Given that an imaginary number is used as a step size to compute the first derivative in the CS derivative method, the calculation stability and accuracy are enhanced with elimination of the subtractive cancellation error, which is commonly encountered when using the traditional finite difference method. The proposed approach unifies the CS approximation and SORM to enhance the estimation of the probability and its sensitivity. The sensitivity analysis facilitates the use of gradient-based optimization algorithms in the RBDO framework. The proposed RBDO/CS–SORM method is tested on structural optimization problems with a range of statistical variations. The results demonstrate that the performance can be enhanced while satisfying precisely probabilistic constraints, thereby increasing the efficiency and efficacy of the optimal design identification. The numerical optimization results obtained using different optimization approaches are compared to validate this enhancement.

The Complex-Step Derivative Approximation on Matrix Lie Groups

The complex-step derivative approximation is a numerical differentiation technique that can achieve analytical accuracy, to machine precision, with a single function evaluation. In this letter, the complex-step derivative approximation is extended to be compatible with elements of matrix Lie groups. As with the standard complex-step derivative, the method is still able to achieve analytical accuracy, up to machine precision, with a single function evaluation. Compared to a central-difference scheme, the proposed complex-step approach is shown to have superior accuracy. The approach is applied to two different pose estimation problems, and is able to recover the same results as an analytical method when available.

Computation of High-Order Electromagnetic Field Sensitivities With FDTD and the Complex-Step Derivative Approximation

This paper introduces a new approach for the computation of electromagnetic field derivatives, up to any order, with respect to the material and geometric parameters of a given geometry, in a single Finite-Difference Time-Domain (FDTD) simulation. The proposed method is based on embedding the complex-step derivative (CSD) approximation into the standard FDTD update equations. Being finite-difference free, CSD provides accurate derivative approximations even for very small perturbations of the design parameters, unlike finite-difference approximations that are prone to subtractive cancellation errors. The availability of accurate approximations of field derivatives with respect to design parameters enables studies such as sensitivity analysis of multiple objective functions (as derivatives of those can be derived from field derivatives via the chain rule), uncertainty quantification, as well as multi-parametric modeling and optimization of electromagnetic structures. The theory, FDTD implementation and applications of this technique are presented.

Finite Element Analysis of High Tensile Strength Steel Sheet by Using Complex Step Derivative Approximations

The framework for the complex step derivative approximations (hereafter CDSA) to calculate the consistent tangent moduli is studied. The present methods is one of the most effective methods to implement any material constitutive equations to the commercial finite element codes and does not suffer from calculation conditions and errors. In order to confirm the efficiency of CDSA, we developed the user subroutine code based on the CDSA using associative J2 flow rules with general nonlinear isotropic hardening rules that is commonly and widely utilized in commercial finite element codes. In this study, the user material subroutine ‘Hypela2’ of MSC.Marc (ver.2013.0.0) was utilized. The finite element calculation result by the proposal method shows a good agreement with the corresponding result by the MSC.Marc default setting. Also we apply the Yoshida-Uemori back stress model to the CDSA and evaluate this new technique to predict the deformation behavior of high tensile strength steel sheet.

2806 複素数階偏微分法による高張力鋼板のプレス成形シミュレーション

本報告では,複雑な弾塑性変形挙動を汎用静的陰解有限要素法(以下,静的陰解汎用FEM)で解析する際に使用する高精度構成方程式を比較的安易かつ高精度に導入することを可能とする複素数階微分法について検討した。本解析手法は,複素数とFortran言語が有する特性を利用することで,あらゆる構成方程式を静的陰解汎用FEMに導入することができることが確認できた.また,上記手法を高精度構成方程式に適用し,高張力鋼板の塑性加工解析を行ったので,それについても報告する.

OS0410 複素数階微分近似法を用いたゴムの粘弾性構成則の実装

OS0410 複素数階微分近似法を用いたゴムの粘弾性構成則の実装

First and Second Complex-step Derivative Approximation of Strain Energy Function and its application to Hyperelastic Model

First and Second Complex-step Derivative Approximation of Strain Energy Function and its application to Hyperelastic Model

Complex differentiation tools for geophysical inversion

I propose a method for derivative approximation, virtually unknown in the geophysical literature but is rapidly gaining recognition in other areas of computational science. The technique, called the complex-step derivative (CSD), is based on the theory of complex variables and approximates first-order derivatives of real-valued functions that are analytic in the complex plane. I extend the CSD technique to second-order derivatives, while preserving the robustness of the first-order formula. Thus the formulas together provide a complete differentiation system (referred to as semiautomatic differentiation, or SD) that allows efficient and accurate approximation of gradients, Jacobians and Hessians, as well as 2D and 3D partial and cross-partial spatial derivatives. Performance evaluation tests indicate that in comparison with ordinary finite-difference schemes (FD), the SD scheme is six to eight orders of magnitude more accurate, numerically highly stable, and step-size insensitive, which are major advantages over FD. The method shares with FD its attractive feature of ease of implementation. The SD scheme is implemented in a MATLAB toolkit. The validity of the CSD method depends critically on the requirement that the target function of differentiation be analytic in the complex plane. Therefore, prior to using the CSD method, one must ensure that all functions in the complex library are defined so that they satisfy this requirement. Specialized complex-function libraries that resolve this and other technical issues for CSD applications are available in the public domain.

On the extension of the complex-step derivative technique to pseudospectral algorithms

The complex-step derivative (CSD) technique is a convenient and highly accurate strategy to perform a linearized “perturbation” analysis to determine a “directional derivative” via a minor modification of an existing nonlinear simulation code. The technique has previously been applied to nonlinear simulation codes (such as finite-element codes) which employ real arithmetic only. The present note examines the suitability of this technique for extension to efficient pseudospectral simulation codes which nominally use the fast Fourier transform (FFT) to convert back and forth between the physical and transformed representations of the system. It is found that, if used carefully, this extension retains the remarkable accuracy of the CSD approach. However, to perform this extension without sacrificing this accuracy, particular care must be exercised; specifically, the state (real) and perturbation (imaginary) components of the complexified system must be transformed separately and arranged in such a manner that they are kept distinct during the process of differentiation in the transformed space in order to avoid the linear combination of the large and small quantities in the analysis. It is shown that this is relatively straightforward to implement even in complicated nonlinear simulation codes, thus positioning the CSD approach as an attractive and relatively simple alternative to hand coding a perturbation (a.k.a. “tangent linear”) code for determining the directional derivative even when pseudospectral algorithms are employed.